C₂³×#2471 #72300200

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂³×#2471

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×#392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

130173 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 26 points:

(8,10)(13,17)(15,18)(22,26)(24,25);(7,8)(9,10)(19,20,23,21)(22,24,26,25);(11,12,14,16)(13,15,17,18)(19,21,23,20)(22,24,26,25);(11,13)(12,15)(14,17)(16,18)(19,22,23,26)(20,24,21,25);(3,4);(5,6);(1,2)(3,4)(5,6)

Conjugacy classes & character table

304 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂³×#35692, C₂³×#35352, C₂³×#51889, …
maximal normal subgroup of	C₂³×#35692, C₂³×#35352, C₂³×#51889, …
maximal quotient of	C₂³×#14905, C₂³×#15368, C₂³×#15669

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((8,10)(13,17)(15,18)(22,26)(24,25), (7,8)(9,10)(19,20,23,21)(22,24,26,25), (11,12,14,16)(13,15,17,18)(19,21,23,20)(22,24,26,25), (11,13)(12,15)(14,17)(16,18)(19,22,23,26)(20,24,21,25), (3,4), (5,6), (1,2)(3,4)(5,6));

C23×#2471 #72300200